On Galois groups of prime degree polynomials with complex roots

Oz Ben-Shimol

Algebra and discrete mathematics, no. 2 (2009), pp. 99-107

Résumé

Let $f$ be an irreducible polynomial of prime degree $p\geq 5$ over $\mathbb{Q}$, with precisely $k$ pairs of complex roots. Using a result of Jens Höchsmann (1999), we show that if $p\geq 4k+1$ then $\rm{Gal}(f/\mathbb{Q})$ is isomorphic to $A_{p}$ or $S_{p}$. This improves the algorithm for computing the Galois group of an irreducible polynomial of prime degree, introduced by A. Bialostocki and T. Shaska. If such a polynomial $f$ is solvable by radicals then its Galois group is a Frobenius group of degree p. Conversely, any Frobenius group of degree p and of even order, can be realized as the Galois group of an irreducible polynomial of degree $p$ over $\mathbb{Q}$ having complex roots.